Analysis of fractional Euler-Bernoulli bending beams using Green’s function method

Abstract

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green’s function for the highest-order term 1<α≤2, which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green’s function kernel. The theory and properties of fractional Green’s function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam’s characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number.

Keywords

• Fractional Green’s function
• Fractional beams
• Beams fractional deflection
• Fractional boundary value problems